Questions: Find f'(x) and simplify. f(x)=x^4 ln x Which of the following shows the correct application of the product rule? A. (x^4)(1/x)+(ln x)(4 x^3) B. (1/x)(4 x^3) C. (x^4)(4 x^3)+(ln x)(1/x) D. (x^4)(1/x)-(ln x)(4 x^3)

Solution Steps

To find the derivative \( f^{\prime}(x) \) of the function \( f(x) = x^4 \ln x \), we will use the product rule. The product rule states that if you have a function \( f(x) = u(x) \cdot v(x) \), then the derivative \( f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \). Here, \( u(x) = x^4 \) and \( v(x) = \ln x \). We will find the derivatives \( u^{\prime}(x) \) and \( v^{\prime}(x) \), and then apply the product rule.

Given the function \( f(x) = x^4 \ln x \), we need to find its derivative using the product rule. The product rule states that if \( f(x) = u(x) \cdot v(x) \), then \( f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \).

Let \( u(x) = x^4 \) and \( v(x) = \ln x \).

• The derivative of \( u(x) \) is \( u^{\prime}(x) = 4x^3 \).
• The derivative of \( v(x) \) is \( v^{\prime}(x) = \frac{1}{x} \).

Using the product rule: \[ f^{\prime}(x) = (4x^3) \cdot (\ln x) + (x^4) \cdot \left(\frac{1}{x}\right) \] Simplifying the expression: \[ f^{\prime}(x) = 4x^3 \ln x + x^3 \]

Factor out \( x^3 \) from the expression: \[ f^{\prime}(x) = x^3 (4 \ln x + 1) \]

The correct application of the product rule is option A: \(\left(x^{4}\right)\left(\frac{1}{x}\right)+(\ln x)\left(4 x^{3}\right)\).

Final Answer